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A082105
Array A(n, k) = (k*n)^2 + 4*(k*n) + 1, read by antidiagonals.
7
1, 1, 1, 1, 6, 1, 1, 13, 13, 1, 1, 22, 33, 22, 1, 1, 33, 61, 61, 33, 1, 1, 46, 97, 118, 97, 46, 1, 1, 61, 141, 193, 193, 141, 61, 1, 1, 78, 193, 286, 321, 286, 193, 78, 1, 1, 97, 253, 397, 481, 481, 397, 253, 97, 1, 1, 118, 321, 526, 673, 726, 673, 526, 321, 118, 1
OFFSET
0,5
LINKS
FORMULA
A(n, k) = (k*n)^2 + 4*(k*n) + 1 (square array).
A(n, n) = T(2*n, n) = A082106(n) (main diagonal).
T(n, k) = A(n-k, k) (number triangle).
Sum_{k=0..n} T(n, k) = A082107(n) (diagonal sums).
T(n, n-1) = A028872(n-1), n >= 1.
T(n, n-2) = A082109(n-2), n >= 2.
From G. C. Greubel, Dec 22 2022: (Start)
Sum_{k=0..n} (-1)^k * T(n, k) = ((1+(-1)^n)/2)*A016897(n-1).
T(2*n+1, n+1) = A047673(n+1), n >= 0.
T(n, n-k) = T(n, k). (End)
EXAMPLE
Array, A(n, k), begins as:
1, 1, 1, 1, 1, 1, ... A000012;
1, 6, 13, 22, 33, 46, ... A028872;
1, 13, 33, 61, 97, 141, ... A082109;
1, 22, 61, 118, 193, 286, ... ;
1, 33, 97, 193, 321, 481, ... ;
1, 46, 141, 286, 481, 726, ... ;
Triangle, T(n, k), begins as:
1;
1, 1;
1, 6, 1;
1, 13, 13, 1;
1, 22, 33, 22, 1;
1, 33, 61, 61, 33, 1;
1, 46, 97, 118, 97, 46, 1;
1, 61, 141, 193, 193, 141, 61, 1;
1, 78, 193, 286, 321, 286, 193, 78, 1;
MATHEMATICA
T[n_, k_]:= (k*(n-k))^2 + 4*(k*(n-k)) + 1;
Table[T[n, k], {n, 0, 13}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 22 2022 *)
PROG
(Magma) [(k*(n-k))^2 + 4*(k*(n-k)) + 1: k in [0..n], n in [0..13]]; // G. C. Greubel, Dec 22 2022
(SageMath)
def A082105(n, k): return (k*(n-k))^2 + 4*(k*(n-k)) + 1
flatten([[A082105(n, k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Dec 22 2022
KEYWORD
easy,nonn,tabl
AUTHOR
Paul Barry, Apr 03 2003
STATUS
approved